Question: For any function $f : A → B$ define an explicit isomorphism between A and the graph $Γ_f ⊂ A×B$ the subset defined by the property that for each $a ∈ A$ there is exactly one pair $(a, b) ∈ Γ_f$ whose first coordinate is $a$
Would the function $$g: A \rightarrow \Gamma_f$$ $$x\mapsto(x,x)$$ work as an isomorphism in this case?
Part two: Define a natural function $Γ_f → B$. Is it necessarily injective? Is it necessarily surjective?
The function I have chosen to define is $$h:(a,b)\mapsto b$$
and it seems to me that it need not be the case that the map is injective given that for $ a_1,a_2 \in A$ and $(a_1,b_1),(a_2,b_2) \in \Gamma_f$ it could be the case that $b_1=b_2$ s.t. $$(a_1,b_1,)\neq(a_2,b_2)$$ $$h((a_1,b_1))=b_1=b_2=h((a_2,b_2))$$
and similarly I don't see why the map need be surjective given that each point in $\Gamma_f$, $(a,b)$ could map to a single $b\in B$
Regarding part one, no, it wouldn't. One simple reason for this is, $g$ takes an element of $a$ and returns element of $A \times A$, not element of $A \times B$, moreover it doesn't depend on the original function $f$.